Discover long patterns

Question

Given a sorted list of numbers, I would like to find the longest subsequence where the differences between successive elements are geometrically increasing. So if the list is <

Answer 1

I think this task is related with not so long ago posted Longest equally-spaced subsequence. I've just modified my algorithm in Python a little bit:

from math import sqrt

def add_precalc(precalc, end, (a, k), count, res, N):
    if end + a * k ** res[1]["count"] > N: return

    x = end + a * k ** count
    if x > N or x < 0: return

    if precalc[x] is None: return

    if (a, k) not in precalc[x]:
        precalc[x][(a, k)] = count

    return

def factors(n):
    res = []
    for x in range(1, int(sqrt(n)) + 1):
        if n % x == 0:
            y = n / x
            res.append((x, y))
            res.append((y, x))
    return res

def work(input):
    precalc = [None] * (max(input) + 1)
    for x in input: precalc[x] = {}

    N = max(input)

    res = ((0, 0), {"end":0, "count":0})
    for i, x in enumerate(input):
        for y in input[i::-1]:
            for a, k in factors(x - y):
                if (a, k) in precalc[x]: continue
                add_precalc(precalc, x, (a, k), 2, res, N)

        for step, count in precalc[x].iteritems():
            count += 1
            if count > res[1]["count"]: res = (step, {"end":x, "count":count})
            add_precalc(precalc, x, step, count, res, N)
        precalc[x] = None

    d = [res[1]["end"]]
    for x in range(res[1]["count"] - 1, 0, -1):
        d.append(d[-1] - res[0][0] * res[0][1] ** x)
    d.reverse()
    return d

explanation

• Traversing the array
• For each previous element of the array calculate factors of the difference between current and taken previous element and then precalculate next possible element of the sequence and saving it to precalc array
• So when arriving at element i there're already all possible sequences with element i in the precalc array, so we have to calculate next possible element and save it to precalc.

Currently there's one place in algorithm that could be slow - factorization of each previous number. I think it could be made faster with two optimizations:

• more effective factorization algorithm
• find a way not to see at each element of array, using the fact that array is sorted and there's already a precalculated sequences

Answer 2

Update

I have made an improvement of the algorithm that it takes an average of O(M + N^2) and memory needs of O(M+N). Mainly is the same that the protocol described below, but to calculate the possible factors A,K for ech diference D, I preload a table. This table takes less than a second to be constructed for M=10^7.

I have made a C implementation that takes less than 10minutes to solve N=10^5 diferent random integer elements.

Here is the source code in C: To execute just do: gcc -O3 -o findgeo findgeo.c

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <memory.h>
#include <time.h>

struct Factor {
    int a;
    int k;
    struct Factor *next;
};

struct Factor *factors = 0;
int factorsL=0;

void ConstructFactors(int R) {
    int a,k,C;
    int R2;
    struct Factor *f;
    float seconds;
    clock_t end;
    clock_t start = clock();

    if (factors) free(factors);
    factors = malloc (sizeof(struct Factor) *((R>>1) + 1));
    R2 = R>>1 ;
    for (a=0;a<=R2;a++) {
        factors[a].a= a;
        factors[a].k=1;
        factors[a].next=NULL;
    }
    factorsL=R2+1;
    R2 = floor(sqrt(R));
    for (k=2; k<=R2; k++) {
        a=1;
        C=a*k*(k+1);
        while (C<R) {
            C >>= 1;
            f=malloc(sizeof(struct Factor));
            *f=factors[C];
            factors[C].a=a;
            factors[C].k=k;
            factors[C].next=f;
            a++;
            C=a*k*(k+1);
        }
    }

    end = clock();
    seconds = (float)(end - start) / CLOCKS_PER_SEC;
    printf("Construct Table: %f\n",seconds);
}

void DestructFactors() {
    int i;
    struct Factor *f;
    for (i=0;i<factorsL;i++) {
        while (factors[i].next) {
            f=factors[i].next->next;
            free(factors[i].next);
            factors[i].next=f;
        }
    }
    free(factors);
    factors=NULL;
    factorsL=0;
}

int ipow(int base, int exp)
{
    int result = 1;
    while (exp)
    {
        if (exp & 1)
            result *= base;
        exp >>= 1;
        base *= base;
    }

    return result;
}

void findGeo(int **bestSolution, int *bestSolutionL,int *Arr, int L) {
    int i,j,D;
    int mustExistToBeBetter;
    int R=Arr[L-1]-Arr[0];
    int *possibleSolution;
    int possibleSolutionL=0;
    int exp;
    int NextVal;
    int idx;
    int kMax,aMax;
    float seconds;
    clock_t end;
    clock_t start = clock();


    kMax = floor(sqrt(R));
    aMax = floor(R/2);
    ConstructFactors(R);
    *bestSolutionL=2;
    *bestSolution=malloc(0);

    possibleSolution = malloc(sizeof(int)*(R+1));

    struct Factor *f;
    int *H=malloc(sizeof(int)*(R+1));
    memset(H,0, sizeof(int)*(R+1));
    for (i=0;i<L;i++) {
        H[ Arr[i]-Arr[0] ]=1;
    }
    for (i=0; i<L-2;i++) {
        for (j=i+2; j<L; j++) {
            D=Arr[j]-Arr[i];
            if (D & 1) continue;
            f = factors + (D >>1);
            while (f) {
                idx=Arr[i] + f->a * f->k  - Arr[0];
                if ((f->k <= kMax)&& (f->a<aMax)&&(idx<=R)&&H[idx]) {
                    if (f->k ==1) {
                        mustExistToBeBetter = Arr[i] + f->a * (*bestSolutionL);
                    } else {
                        mustExistToBeBetter = Arr[i] + f->a * f->k * (ipow(f->k,*bestSolutionL) - 1)/(f->k-1);
                    }
                    if (mustExistToBeBetter< Arr[L-1]+1) {
                        idx=  floor(mustExistToBeBetter - Arr[0]);
                    } else {
                        idx = R+1;
                    }
                    if ((idx<=R)&&H[idx]) {
                        possibleSolution[0]=Arr[i];
                        possibleSolution[1]=Arr[i] + f->a*f->k;
                        possibleSolution[2]=Arr[j];
                        possibleSolutionL=3;
                        exp = f->k * f->k * f->k;
                        NextVal = Arr[j] + f->a * exp;
                        idx=NextVal - Arr[0];
                        while ( (idx<=R) && H[idx]) {
                            possibleSolution[possibleSolutionL]=NextVal;
                            possibleSolutionL++;
                            exp = exp * f->k;
                            NextVal = NextVal + f->a * exp;
                            idx=NextVal - Arr[0];
                        }

                        if (possibleSolutionL > *bestSolutionL) {
                            free(*bestSolution);
                            *bestSolution = possibleSolution;
                            possibleSolution = malloc(sizeof(int)*(R+1));
                            *bestSolutionL=possibleSolutionL;
                            kMax= floor( pow (R, 1/ (*bestSolutionL) ));
                            aMax= floor(R /  (*bestSolutionL));
                        }
                    }
                }
                f=f->next;
            }
        }
    }

    if (*bestSolutionL == 2) {
        free(*bestSolution);
        possibleSolutionL=0;
        for (i=0; (i<2)&&(i<L); i++ ) {
            possibleSolution[possibleSolutionL]=Arr[i];
            possibleSolutionL++;
        }
        *bestSolution = possibleSolution;
        *bestSolutionL=possibleSolutionL;
    } else {
        free(possibleSolution);
    }
    DestructFactors();
    free(H);

    end = clock();
    seconds = (float)(end - start) / CLOCKS_PER_SEC;
    printf("findGeo: %f\n",seconds);
}

int compareInt (const void * a, const void * b)
{
    return *(int *)a - *(int *)b;
}

int main(void) {
    int N=100000;
    int R=10000000;
    int *A = malloc(sizeof(int)*N);
    int *Sol;
    int SolL;
    int i;


    int *S=malloc(sizeof(int)*R);
    for (i=0;i<R;i++) S[i]=i+1;

    for (i=0;i<N;i++) {
        int r = rand() % (R-i);
        A[i]=S[r];
        S[r]=S[R-i-1];
    }

    free(S);
    qsort(A,N,sizeof(int),compareInt);

/*
    int step = floor(R/N);
    A[0]=1;
    for (i=1;i<N;i++) {
        A[i]=A[i-1]+step;
    }
*/

    findGeo(&Sol,&SolL,A,N);

    printf("[");
    for (i=0;i<SolL;i++) {
        if (i>0) printf(",");
        printf("%d",Sol[i]);
    }
    printf("]\n");
    printf("Size: %d\n",SolL);

    free(Sol);
    free(A);
    return EXIT_SUCCESS;
}

Demostration

I will try to demonstrate that the algorithm that I proposed is in average for an equally distributed random sequence. I’m not a mathematician and I am not used to do this kind of demonstrations, so please fill free to correct me any error that you can see.

There are 4 indented loops, the two firsts are the N^2 factor. The M is for the calculation of the possible factors table).

The third loop is executed only once in average for each pair. You can see this checking the size of the pre-calculated factors table. It’s size is M when N->inf. So the average steps for each pair is M/M=1.

So the proof happens to check that the forth loop. (The one that traverses the good made sequences is executed less that or equal O(N^2) for all the pairs.

To demonstrate that, I will consider two cases: one where M>>N and other where M ~= N. Where M is the maximum difference of the initial array: M= S(n)-S(1).

For the first case, (M>>N) the probability to find a coincidence is p=N/M. To start a sequence, it must coincide the second and the b+1 element where b is the length of the best sequence until now. So the loop will enter times. And the average length of this series (supposing an infinite series) is . So the total number of times that the loop will be executed is . And this is close to 0 when M>>N. The problem here is when M~=N.

Now lets consider this case where M~=N. Lets consider that b is the best sequence length until now. For the case A=k=1, then the sequence must start before N-b, so the number of sequences will be N-b, and the times that will go for the loop will be a maximum of (N-b)*b.

For A>1 and k=1 we can extrapolate to where d is M/N (the average distance between numbers). If we add for all A’s from 1 to dN/b then we see a top limit of:

For the cases where k>=2, we see that the sequence must start before , So the loop will enter an average of and adding for all As from 1 to dN/k^b, it gives a limit of

Here, the worst case is when b is minimum. Because we are considering minimum series, lets consider a very worst case of b= 2 so the number of passes for the 4th loop for a given k will be less than

.

And if we add all k’s from 2 to infinite will be:

So adding all the passes for k=1 and k>=2, we have a maximum of:

Note that d=M/N=1/p.

So we have two limits, One that goes to infinite when d=1/p=M/N goes to 1 and other that goes to infinite when d goes to infinite. So our limit is the minimum of both, and the worst case is when both equetions cross. So if we solve the equation:

we see that the maximum is when d=1.353

So it is demonstrated that the forth loops will be processed less than 1.55N^2 times in total.

Of course, this is for the average case. For the worst case I am not able to find a way to generate series whose forth loop are higher than O(N^2), and I strongly believe that they does not exist, but I am not a mathematician to prove it.

Old Answer

Here is a solution in average of O((n^2)*cube_root(M)) where M is the difference between the first and last element of the array. And memory requirements of O(M+N).

1.- Construct an array H of length M so that M[i - S[0]]=true if i exists in the initial array and false if it does not exist.

2.- For each pair in the array S[j], S[i] do:

2.1 Check if it can be the first and third elements of a possible solution. To do so, calculate all possible A,K pairs that meet the equation S(i) = S(j) + AK + AK^2. Check this SO question to see how to solve this problem. And check that exist the second element: S[i]+ A*K

2.2 Check also that exist the element one position further that the best solution that we have. For example, if the best solution that we have until now is 4 elements long then check that exist the element A[j] + AK + AK^2 + AK^3 + AK^4

2.3 If 2.1 and 2.2 are true, then iterate how long is this series and set as the bestSolution until now is is longer that the last.

Here is the code in javascript:

function getAKs(A) {
    if (A / 2 != Math.floor(A / 2)) return [];
    var solution = [];
    var i;
    var SR3 = Math.pow(A, 1 / 3);
    for (i = 1; i <= SR3; i++) {
        var B, C;
        C = i;
        B = A / (C * (C + 1));
        if (B == Math.floor(B)) {
            solution.push([B, C]);
        }

        B = i;
        C = (-1 + Math.sqrt(1 + 4 * A / B)) / 2;
        if (C == Math.floor(C)) {
            solution.push([B, C]);
        }
    }

    return solution;
}

function getBestGeometricSequence(S) {
    var i, j, k;

    var bestSolution = [];

    var H = Array(S[S.length-1]-S[0]);
    for (i = 0; i < S.length; i++) H[S[i] - S[0]] = true;

    for (i = 0; i < S.length; i++) {
        for (j = 0; j < i; j++) {
            var PossibleAKs = getAKs(S[i] - S[j]);
            for (k = 0; k < PossibleAKs.length; k++) {
                var A = PossibleAKs[k][0];
                var K = PossibleAKs[k][17];

                var mustExistToBeBetter;
                if (K==1) {
                    mustExistToBeBetter = S[j] + A * bestSolution.length;
                } else {
                    mustExistToBeBetter = S[j] + A * K * (Math.pow(K,bestSolution.length) - 1)/(K-1);
                }

                if ((H[S[j] + A * K - S[0]]) && (H[mustExistToBeBetter - S[0]])) {
                    var possibleSolution=[S[j],S[j] + A * K,S[i]];
                    exp = K * K * K;
                    var NextVal = S[i] + A * exp;
                    while (H[NextVal - S[0]] === true) {
                        possibleSolution.push(NextVal);
                        exp = exp * K;
                        NextVal = NextVal + A * exp;
                    }

                    if (possibleSolution.length > bestSolution.length) {
                        bestSolution = possibleSolution;
                    }
                }
            }
        }
    }
    return bestSolution;
}

//var A= [ 1, 2, 3,5,7, 15, 27, 30,31, 81];
var A=[];
for (i=1;i<=3000;i++) {
    A.push(i);
}
var sol=getBestGeometricSequence(A);

$("#result").html(JSON.stringify(sol));

You can check the code here: http://jsfiddle.net/6yHyR/1/

I maintain the other solution because I believe that it is still better when M is very big compared to N.

Answer 3

Python:

def subseq(a):
    seq = []
    aset = set(a)
    for i, x in enumerate(a):
        # elements after x
        for j, x2 in enumerate(a[i+1:]):
            j += i + 1  # enumerate starts j at 0, we want a[j] = x2
            bk = x2 - x  # b*k (assuming k and k's exponent start at 1)

            # given b*k, bruteforce values of k
            for k in range(1, bk + 1):
                items = [x, x2]  # our subsequence so far
                nextdist = bk * k # what x3 - x2 should look like

                while items[-1] + nextdist in aset:
                    items.append(items[-1] + nextdist)
                    nextdist *= k

                if len(items) > len(seq):
                    seq = items

    return seq

Running time is O(dn^3), where d is the (average?) distance between two elements, and n is of course len(a).

Answer 4

Just to start with something, here is a simple solution in JavaScript:

var input = [0.7, 1, 2, 3, 4, 7, 15, 27, 30, 31, 81], 
    output = [], indexes, values, i, index, value, i_max_length,
    i1, i2, i3, j1, j2, j3, difference12a, difference23a, difference12b, difference23b,
    scale_factor, common_ratio_a, common_ratio_b, common_ratio_c,
    error, EPSILON = 1e-9, common_ratio_is_integer,
    resultDiv = $("#result");

for (i1 = 0; i1 < input.length - 2; ++i1) {
    for (i2 = i1 + 1; i2 < input.length - 1; ++i2) {
        scale_factor = difference12a = input[i2] - input[i1];
        for (i3 = i2 + 1; i3 < input.length; ++i3) {
            difference23a = input[i3] - input[i2];
            common_ratio_1a = difference23a / difference12a;
            common_ratio_2a = Math.round(common_ratio_1a);
            error = Math.abs((common_ratio_2a - common_ratio_1a) / common_ratio_1a);
            common_ratio_is_integer = error < EPSILON;
            if (common_ratio_2a > 1 && common_ratio_is_integer) {
                indexes = [i1, i2, i3];
                j1 = i2;
                j2 = i3
                difference12b = difference23a;
                for (j3 = j2 + 1; j3 < input.length; ++j3) {
                    difference23b = input[j3] - input[j2];
                    common_ratio_1b = difference23b / difference12b;
                    common_ratio_2b = Math.round(common_ratio_1b);
                    error = Math.abs((common_ratio_2b - common_ratio_1b) / common_ratio_1b);
                    common_ratio_is_integer = error < EPSILON;
                    if (common_ratio_is_integer && common_ratio_2a === common_ratio_2b) {
                        indexes.push(j3);
                        j1 = j2;
                        j2 = j3
                        difference12b = difference23b;
                    }
                }
                values = [];
                for (i = 0; i < indexes.length; ++i) {
                    index = indexes[i];
                    value = input[index];
                    values.push(value);
                }
                output.push(values);
            }
        }
    }
}
if (output !== []) {
    i_max_length = 0;
    for (i = 1; i < output.length; ++i) {
        if (output[i_max_length].length < output[i].length)
            i_max_length = i;
    }
    for (i = 0; i < output.length; ++i) {
        if (output[i_max_length].length == output[i].length)
            resultDiv.append("<p>[" + output[i] + "]</p>");
    }
}

Output:

[1, 3, 7, 15, 31]

I find the first three items of every subsequence candidate, calculate the scale factor and the common ratio from them, and if the common ratio is integer, then I iterate over the remaining elements after the third one, and add those to the subsequence, which fit into the geometric progression defined by the first three items. As a last step, I select the sebsequence/s which has/have the largest length.

Answer 5

Here is my solution in Javascript. It should be close to O(n^2) except may be in some pathological cases.

function bsearch(Arr,Val, left,right) {
    if (left == right) return left;
    var m=Math.floor((left + right) /2);
    if (Val <= Arr[m]) {
        return bsearch(Arr,Val,left,m);
    } else {
        return bsearch(Arr,Val,m+1,right);
    }
}

function findLongestGeometricSequence(S) {
    var bestSolution=[];

    var i,j,k;
    var H={};
    for (i=0;i<S.length;i++) H[S[i]]=true;

    for (i=0;i<S.length;i++) {
        for (j=0;j<i;j++) {
            for (k=j+1;k<i;) {
                var possibleSolution=[S[j],S[k],S[i]];

                var K = (S[i] - S[k]) / (S[k] - S[j]);
                var A = (S[k] - S[j]) * (S[k] - S[j]) / (S[i] - S[k]); 

                if ((Math.floor(K) == K) && (Math.floor(A)==A)) {
                    exp= K*K*K;
                    var NextVal= S[i] + A * exp;
                    while (H[NextVal] === true) {
                        possibleSolution.push(NextVal);
                        exp = exp * K;
                        NextVal= NextVal + A * exp;                
                    }

                    if (possibleSolution.length > bestSolution.length)
                        bestSolution=possibleSolution;

                    K--;
                } else {
                    K=Math.floor(K);
                }

                if (K>0) {
                    var NextPossibleMidValue= (S[i] + K*S[j]) / (K +1); 
                    k++;
                    if (S[k]<NextPossibleMidValue) {
                        k=bsearch(S,NextPossibleMidValue, k+1, i);
                    }
                } else {
                    k=i;
                }
            }        
        }
    }
    return bestSolution;
}


function Run() {
    var MyS= [0.7, 1, 2, 3, 4, 5,6,7, 15, 27, 30,31, 81];

    var sol = findLongestGeometricSequence(MyS); 

    alert(JSON.stringify(sol));
}

Small Explanation

If we take 3 numbers of the array S(j) < S(k) < S(i) then you can calculate a and k so that: S(k) = S(j) + a*k and S(i) = S(k) + a*k^2 (2 equations and 2 incognits). With that in mind, you can check if exist a number in the array that is S(next) = S(i) + a*k^3. If that is the case, then continue checknng for S(next2) = S(next) + a*k^4 and so on.

This would be a O(n^3) solution, but you can hava advantage that k must be integer in order to limit the S(k) points selected.

In case that a is known, then you can calculate a(k) and you need to check only one number in the third loop, so this case will be clearly a O(n^2).

Answer 6

In fact it is exactly the same question as Longest equally-spaced subsequence, you just have to consider the logarithm of your data. If the sequence is a, ak, ak^2, ak^3, the logarithmique value is ln(a), ln(a) + ln(k), ln(a)+2ln(k), ln(a)+3ln(k), so it is equally spaced. The opposite is of course true. There is a lot of different code in the question above.

I don't think the special case a=1 can be resolved more efficiently than an adaptation from an algorithm above.